Francesco Cavazzani

Enumerative questions in algebraic geometry go all the way back to the second
half of the 19th century, and before; even though evolving in format and langua-
ge, the main underlying idea has always been been finding a compact moduli
space for the problem, and doing intersection theory on it. About twisted cu-
bics, the most common moduli spaces (Hilbert schemes, stable maps, and more)
partially solved all the question Schubert raised (and answered, using not enti-
rely understood methods) in 1879. In this talk, we will construct a new moduli
space of twisted cubics, obtained compactifying $PGL_4$ to the so-called space of
complete collineations and then taking the GIT quotient by $PGL_2$. The space
so obtained is very symmetric; in fact, following the theory of homogeneous
spaces, it is possible to link plenty of geometric properties of this space to re-
presentation theoretic properties of $PGL_4$ and $PGL_2$. In this way, intersection
theory on this space becomes just a combinatorial problem involving generating
functions, partition functions, and interpolation; the number 56960 of twisted
cubics tangent to 12 given planes is just the integral of a piecewise polynomial
over a 3 dimensional region. This is a work in progress towards my PhD thesis.